The Gunners headed to Lancashire knowing that three points would be enough to capitalise on some favourable results yesterday and lift them into a Champions League spot.
For so long it looked like being a goalless draw in a match of few clear-cut chances - the best of which fell Burnley's way - but James Tarkowski's shove on Aaron Ramsey in added time allowed Sanchez to convert from 12 yards.
Arsenal saw far more of the ball in advanced positions in the second period, though, with Ben Mee twice having to put his body on the line to thwart the visitors.
Pope was called into action for the first time shortly before the hour to keep out Lacazette's tame attempt, while Ramsey prodded wide from a tight angle against a side boasting the third-best defensive record in the division.
Arsenal felt that they should have had a penalty when Bellerin was taken down by Brady inside the box, with referee Lee Mason instead waving play to continue, soon after Gudmundsson's powerful on-target drive was gathered by Cech.
The Gunners required late goals in both league fixtures against Burnley last term and were pressing hard for one here, seeing Wilshere's strike deflect wide and another from Ramsey sail over.
Just when it looked as though both teams would be denied a place in the top four, Tarkowski's needless push on Ramsey gifted Sanchez the chance to snatch all three points, which he made the most of by tucking the penalty out of Pope's reach.
Burnley (4-5-1): Pope; Lowton, Tarkowski, Mee, Ward; Gudmundsson, Cork, Hendrick, Defour (Wood 78'), Brady; Barnes
Arsenal (3-4-2-1): Cech; Koscielny, Mustafi, Monreal; Bellerin, Ramsey, Xhaka, Kolasinac; Iwobi (Wilshere 67'), Sanchez; Lacazette (Welbeck 78')